Vandermonde sets, hyperovals and Niho bent functions

Kanat Abdukhalikov; Duy Ho

doi:10.3934/amc.2021048

Article Contents

2023, Volume 17, Issue 5: 1235-1250. Doi: 10.3934/amc.2021048

This issue Previous Article On the complexity of solving generic overdetermined bilinear systems Next Article A survey on functional encryption

Vandermonde sets, hyperovals and Niho bent functions

Kanat Abdukhalikov^1, , and
Duy Ho^2,

1.
UAE University, PO Box 15551, Al Ain, UAE, and Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan
2.
UAE University, PO Box 15551, Al Ain, UAE

^* Corresponding author: Kanat Abdukhalikov
^* Corresponding author: Kanat Abdukhalikov

Received: June 2021

Revised: September 2021

Early access: October 2021

Published: October 2023

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Abstract

We consider relationships between Vandermonde sets and hyperovals. Hyperovals are Vandermonde sets, but in general, Vandermonde sets are not hyperovals. We give necessary and sufficient conditions for a Vandermonde set to be a hyperoval in terms of power sums. Therefore, we provide purely algebraic criteria for the existence of hyperovals. Furthermore, we give necessary and sufficient conditions for the existence of hyperovals in terms of $ g $-functions, which can be considered as an analog of Glynn's Theorem for $ o $-polynomials. We also get some important applications to Niho bent functions.

Keywords:

Mathematics Subject Classification: Primary: 51E21, 94A60; Secondary: 51E15, 51E23.

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Table 1. Set $ \mathcal{D} $ in small fields

$ q $	Elements in $ \mathcal{D} $	$ \mid \mathcal{D} \mid $
4	1	1
8	1, 3, 5	3
16	1, 3, 5, 7, 9, 11, 13, 37	8
32	1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 69, 73, 77, 85, 89,147	21
64	1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61,133,137,141,145,149,153,157,165,169,173,177,181,185,275,281,283,291,297,299,307,313,409,425,661	55
128	1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99,101,103,105,107,109,111,113,115,117,119,121,123,125,261,265,269,273,277,281,285,289,293,297,301,305,309,313,317,325,329,333,337,341,345,349,353,357,361,365,369,373,377,529,531,537,539,547,553,555,561,563,569,571,579,585,587,593,595,601,603,611,617,619,625,627,633,785,793,809,817,825,841,849,857,873,881, 1093, 1095, 1107, 1109, 1111, 1123, 1125, 1127, 1139, 1141, 1301, 1317, 1333, 1365, 1381, 1587, 1619, 2341, 2349, 2381, 2405	147

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